Spoiler Alerts

Useful constants: $c = 3 \times 10^{10}~\mathrm{cm/s}$ ; $1~\mathrm{pc} \approx 3.26~\mathrm{ly}$ ; $1~\mathrm{yr} \approx 3.15 \times 10^7~\mathrm{s}$ ; Sun–Galactic-center orbital period $\approx 2.3 \times 10^8~\mathrm{yr}$ .

For each problem, be explicit about what was measured versus what was inferred, and name the model that bridges the two.

Conceptual

1. ⭐ The measurement ceiling. A student lists luminosity, temperature, mass, and distance as the four things astronomers directly measure. Correct this statement. What can astronomers directly measure, and how do we actually determine each item on the student’s list?

2. ⭐⭐ Standard candle logic. Explain why a standard candle must be both intrinsically luminous enough and reliably calibrated. What goes wrong if either condition fails?

3. ⭐⭐ Observable vs. model-dependent. A news article claims “Astronomers measured the temperature of TRAPPIST-1e to be 255 K.” Critique this statement. What was actually measured, and what model was used to arrive at the temperature?

4. ⭐⭐ Empirical vs. physical. What is the difference between an empirical law and a physical law? Give one example from Lecture 1 where an empirical pattern is explained by a physical mechanism, and explain why that distinction matters.

Calculation

5. ⭐⭐ 21-cm hydrogen line. The 21-cm line traces cold neutral hydrogen.

(a) Convert 21 cm to meters and to nanometers.
(b) Compute the frequency using $c = \lambda \nu$ with $c = 3 \times 10^{10}$ cm/s.
(c) Show a unit check that gives s $^{-1}$ .
(d) Limiting case: X-rays have $\lambda \sim 10^{-8}$ cm. Without calculating, predict whether their frequency is higher or lower than 21-cm radiation, and by roughly how many orders of magnitude.

6. ⭐⭐ Inverse-square distance. Two identical Cepheid variables are observed. Star A has flux 25 times greater than Star B.

(a) Derive the distance ratio $d_B/d_A$ .
(b) If Star A is at 1 kpc, what is Star B’s distance in kpc?
(c) Convert Star B’s distance to light-years ( $1~\mathrm{pc} \approx 3.26$ ly).

7. ⭐⭐ Light-travel across the galaxy. The Milky Way is about 100,000 light-years across.

(a) How long does light take to cross it? Express in years.
(b) Convert this to seconds using $1~\mathrm{yr} \approx 3.15 \times 10^7$ s.
(c) Sanity check: the Sun orbits the Galactic center in ~230 million years. Roughly how many galactic orbits occur during one light-crossing time?

Synthesis

8. ⭐⭐ The inference chain. For each scenario, identify (a) what is directly measured, (b) what model connects measurement to inference, and (c) what is inferred:

(i) Determining the composition of the Sun
(ii) Determining the mass of a planet orbiting another star
(iii) Determining the age of a star cluster

9. ⭐⭐ (Challenge) The measurement-inference web. A star is reported to have mass $\sim 15\,M_\odot$ , radius $\sim 900\,R_\odot$ , luminosity $\sim 10^5\,L_\odot$ , and distance $\sim 650$ light-years.

(a) For each property, identify which observable(s) are needed to infer it.
(b) Which properties depend on knowing the others first?
(c) If the distance were revised upward by 20%, which inferences would change, and how?